Ultralight bosons exist in various proposed extensions to the Standard Model, which can form condensates around rapidly rotating black holes through a process called superradiance. These boson clouds have many interesting observational consequences, such as the continuous emission of monochromatic gravitational waves. In this talk, I will describe the dynamics of the system when it is part of a binary black hole. I will show that the presence of a binary companion greatly enriches the evolution of the boson clouds, most remarkably through the existence of resonant transitions between growing and decaying modes of the clouds. Finally, I will sketch some phenomenological consequences, both for the gravitational waves emitted by the clouds and the finite-size effects imprinted in the waveforms of the binary signal.